I need to find the following limit of a function of complex variable: <mtable columnalign="right

Ryan Reynolds

Ryan Reynolds

Answered question

2022-05-24

I need to find the following limit of a function of complex variable:
lim z i z + i 3 2 i 3 z i

Answer & Explanation

Madisyn Avery

Madisyn Avery

Beginner2022-05-25Added 12 answers

Here is a computation of the principal value. There are 6 possible values of the square and cube roots, but I'll let myself be bashed for ignoring them.
lim z i z + i 3 2 i 3 z i = lim w + i i w + i + i 3 2 i 3 w + i i = lim w 0 w + 2 i 3 2 i 3 w + i i = lim w 0 2 i 3 1 + w / 2 i 3 3 2 i 3 i 1 + w / i i = lim w 0 2 i 3 ( 1 + w / 2 i 3 3 1 ) i ( 1 + w / i 1 ) = 2 i 3 i lim w 0 1 + w / ( 3 2 i 3 ) + O ( w 2 ) 1 1 + w / ( 2 i ) + O ( w 2 ) 1 ) = 2 i 3 i lim w 0 w / ( 3 2 i 3 ) + O ( w 2 ) w / ( 2 i ) + O ( w 2 ) = 2 i 3 i lim w 0 ( 2 i 3 2 i 3 + O ( w ) ) = 2 3

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